Question

The condition $f\left(x\right)=x^3+p{x}^2+qx+r,\left(x\in R\right)$ to have no extreme value, is

• A. $p^2<3q$
• B. $2p^2<q$
• C. $p^2<\left(\frac{1}{4}\right)q$
• D. $p^2>3q$

Answer 1

The correct option is A

$p^2<3q$

Explanation for the correct option:

Finding the condition:

Given Equation $f\left(x\right)=x^3+p{x}^2+qx+r,\left(x\in R\right)$

To have no extreme value, $f\text{'}\left(x\right)\ne 0$

$f\text{'}\left(x\right)=3x^2+2px+q\ne 0$

And also, the discriminant should be less than $0$,

Hence, $D<0$

$$\begin{gathered} \Rightarrow 4p^2–\left(4\right)\times \left(3\right)\times \left(q\right)<0 \\ \Rightarrow 4p^2–12q<0 \\ \Rightarrow p^2<3q \end{gathered}$$

Hence, option (A) is the correct answer.